Integrand size = 11, antiderivative size = 132 \[ \int x^{10} (a+b x)^{10} \, dx=\frac {a^{10} x^{11}}{11}+\frac {5}{6} a^9 b x^{12}+\frac {45}{13} a^8 b^2 x^{13}+\frac {60}{7} a^7 b^3 x^{14}+14 a^6 b^4 x^{15}+\frac {63}{4} a^5 b^5 x^{16}+\frac {210}{17} a^4 b^6 x^{17}+\frac {20}{3} a^3 b^7 x^{18}+\frac {45}{19} a^2 b^8 x^{19}+\frac {1}{2} a b^9 x^{20}+\frac {b^{10} x^{21}}{21} \]
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Time = 0.04 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int x^{10} (a+b x)^{10} \, dx=\frac {a^{10} x^{11}}{11}+\frac {5}{6} a^9 b x^{12}+\frac {45}{13} a^8 b^2 x^{13}+\frac {60}{7} a^7 b^3 x^{14}+14 a^6 b^4 x^{15}+\frac {63}{4} a^5 b^5 x^{16}+\frac {210}{17} a^4 b^6 x^{17}+\frac {20}{3} a^3 b^7 x^{18}+\frac {45}{19} a^2 b^8 x^{19}+\frac {1}{2} a b^9 x^{20}+\frac {b^{10} x^{21}}{21} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (a^{10} x^{10}+10 a^9 b x^{11}+45 a^8 b^2 x^{12}+120 a^7 b^3 x^{13}+210 a^6 b^4 x^{14}+252 a^5 b^5 x^{15}+210 a^4 b^6 x^{16}+120 a^3 b^7 x^{17}+45 a^2 b^8 x^{18}+10 a b^9 x^{19}+b^{10} x^{20}\right ) \, dx \\ & = \frac {a^{10} x^{11}}{11}+\frac {5}{6} a^9 b x^{12}+\frac {45}{13} a^8 b^2 x^{13}+\frac {60}{7} a^7 b^3 x^{14}+14 a^6 b^4 x^{15}+\frac {63}{4} a^5 b^5 x^{16}+\frac {210}{17} a^4 b^6 x^{17}+\frac {20}{3} a^3 b^7 x^{18}+\frac {45}{19} a^2 b^8 x^{19}+\frac {1}{2} a b^9 x^{20}+\frac {b^{10} x^{21}}{21} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00 \[ \int x^{10} (a+b x)^{10} \, dx=\frac {a^{10} x^{11}}{11}+\frac {5}{6} a^9 b x^{12}+\frac {45}{13} a^8 b^2 x^{13}+\frac {60}{7} a^7 b^3 x^{14}+14 a^6 b^4 x^{15}+\frac {63}{4} a^5 b^5 x^{16}+\frac {210}{17} a^4 b^6 x^{17}+\frac {20}{3} a^3 b^7 x^{18}+\frac {45}{19} a^2 b^8 x^{19}+\frac {1}{2} a b^9 x^{20}+\frac {b^{10} x^{21}}{21} \]
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Time = 0.18 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.86
method | result | size |
gosper | \(\frac {1}{11} a^{10} x^{11}+\frac {5}{6} a^{9} b \,x^{12}+\frac {45}{13} a^{8} b^{2} x^{13}+\frac {60}{7} a^{7} b^{3} x^{14}+14 a^{6} b^{4} x^{15}+\frac {63}{4} a^{5} b^{5} x^{16}+\frac {210}{17} a^{4} b^{6} x^{17}+\frac {20}{3} a^{3} b^{7} x^{18}+\frac {45}{19} a^{2} b^{8} x^{19}+\frac {1}{2} a \,b^{9} x^{20}+\frac {1}{21} b^{10} x^{21}\) | \(113\) |
default | \(\frac {1}{11} a^{10} x^{11}+\frac {5}{6} a^{9} b \,x^{12}+\frac {45}{13} a^{8} b^{2} x^{13}+\frac {60}{7} a^{7} b^{3} x^{14}+14 a^{6} b^{4} x^{15}+\frac {63}{4} a^{5} b^{5} x^{16}+\frac {210}{17} a^{4} b^{6} x^{17}+\frac {20}{3} a^{3} b^{7} x^{18}+\frac {45}{19} a^{2} b^{8} x^{19}+\frac {1}{2} a \,b^{9} x^{20}+\frac {1}{21} b^{10} x^{21}\) | \(113\) |
norman | \(\frac {1}{11} a^{10} x^{11}+\frac {5}{6} a^{9} b \,x^{12}+\frac {45}{13} a^{8} b^{2} x^{13}+\frac {60}{7} a^{7} b^{3} x^{14}+14 a^{6} b^{4} x^{15}+\frac {63}{4} a^{5} b^{5} x^{16}+\frac {210}{17} a^{4} b^{6} x^{17}+\frac {20}{3} a^{3} b^{7} x^{18}+\frac {45}{19} a^{2} b^{8} x^{19}+\frac {1}{2} a \,b^{9} x^{20}+\frac {1}{21} b^{10} x^{21}\) | \(113\) |
risch | \(\frac {1}{11} a^{10} x^{11}+\frac {5}{6} a^{9} b \,x^{12}+\frac {45}{13} a^{8} b^{2} x^{13}+\frac {60}{7} a^{7} b^{3} x^{14}+14 a^{6} b^{4} x^{15}+\frac {63}{4} a^{5} b^{5} x^{16}+\frac {210}{17} a^{4} b^{6} x^{17}+\frac {20}{3} a^{3} b^{7} x^{18}+\frac {45}{19} a^{2} b^{8} x^{19}+\frac {1}{2} a \,b^{9} x^{20}+\frac {1}{21} b^{10} x^{21}\) | \(113\) |
parallelrisch | \(\frac {1}{11} a^{10} x^{11}+\frac {5}{6} a^{9} b \,x^{12}+\frac {45}{13} a^{8} b^{2} x^{13}+\frac {60}{7} a^{7} b^{3} x^{14}+14 a^{6} b^{4} x^{15}+\frac {63}{4} a^{5} b^{5} x^{16}+\frac {210}{17} a^{4} b^{6} x^{17}+\frac {20}{3} a^{3} b^{7} x^{18}+\frac {45}{19} a^{2} b^{8} x^{19}+\frac {1}{2} a \,b^{9} x^{20}+\frac {1}{21} b^{10} x^{21}\) | \(113\) |
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Time = 0.22 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85 \[ \int x^{10} (a+b x)^{10} \, dx=\frac {1}{21} \, b^{10} x^{21} + \frac {1}{2} \, a b^{9} x^{20} + \frac {45}{19} \, a^{2} b^{8} x^{19} + \frac {20}{3} \, a^{3} b^{7} x^{18} + \frac {210}{17} \, a^{4} b^{6} x^{17} + \frac {63}{4} \, a^{5} b^{5} x^{16} + 14 \, a^{6} b^{4} x^{15} + \frac {60}{7} \, a^{7} b^{3} x^{14} + \frac {45}{13} \, a^{8} b^{2} x^{13} + \frac {5}{6} \, a^{9} b x^{12} + \frac {1}{11} \, a^{10} x^{11} \]
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Time = 0.03 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.99 \[ \int x^{10} (a+b x)^{10} \, dx=\frac {a^{10} x^{11}}{11} + \frac {5 a^{9} b x^{12}}{6} + \frac {45 a^{8} b^{2} x^{13}}{13} + \frac {60 a^{7} b^{3} x^{14}}{7} + 14 a^{6} b^{4} x^{15} + \frac {63 a^{5} b^{5} x^{16}}{4} + \frac {210 a^{4} b^{6} x^{17}}{17} + \frac {20 a^{3} b^{7} x^{18}}{3} + \frac {45 a^{2} b^{8} x^{19}}{19} + \frac {a b^{9} x^{20}}{2} + \frac {b^{10} x^{21}}{21} \]
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Time = 0.22 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85 \[ \int x^{10} (a+b x)^{10} \, dx=\frac {1}{21} \, b^{10} x^{21} + \frac {1}{2} \, a b^{9} x^{20} + \frac {45}{19} \, a^{2} b^{8} x^{19} + \frac {20}{3} \, a^{3} b^{7} x^{18} + \frac {210}{17} \, a^{4} b^{6} x^{17} + \frac {63}{4} \, a^{5} b^{5} x^{16} + 14 \, a^{6} b^{4} x^{15} + \frac {60}{7} \, a^{7} b^{3} x^{14} + \frac {45}{13} \, a^{8} b^{2} x^{13} + \frac {5}{6} \, a^{9} b x^{12} + \frac {1}{11} \, a^{10} x^{11} \]
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Time = 0.30 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85 \[ \int x^{10} (a+b x)^{10} \, dx=\frac {1}{21} \, b^{10} x^{21} + \frac {1}{2} \, a b^{9} x^{20} + \frac {45}{19} \, a^{2} b^{8} x^{19} + \frac {20}{3} \, a^{3} b^{7} x^{18} + \frac {210}{17} \, a^{4} b^{6} x^{17} + \frac {63}{4} \, a^{5} b^{5} x^{16} + 14 \, a^{6} b^{4} x^{15} + \frac {60}{7} \, a^{7} b^{3} x^{14} + \frac {45}{13} \, a^{8} b^{2} x^{13} + \frac {5}{6} \, a^{9} b x^{12} + \frac {1}{11} \, a^{10} x^{11} \]
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Time = 0.05 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85 \[ \int x^{10} (a+b x)^{10} \, dx=\frac {a^{10}\,x^{11}}{11}+\frac {5\,a^9\,b\,x^{12}}{6}+\frac {45\,a^8\,b^2\,x^{13}}{13}+\frac {60\,a^7\,b^3\,x^{14}}{7}+14\,a^6\,b^4\,x^{15}+\frac {63\,a^5\,b^5\,x^{16}}{4}+\frac {210\,a^4\,b^6\,x^{17}}{17}+\frac {20\,a^3\,b^7\,x^{18}}{3}+\frac {45\,a^2\,b^8\,x^{19}}{19}+\frac {a\,b^9\,x^{20}}{2}+\frac {b^{10}\,x^{21}}{21} \]
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